Analisis de Perturbaciones en Sistemas Interconectados de Potencia

8/18/2019 Analisis de Perturbaciones en Sistemas Interconectados de Potencia

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By

Mr. Richard Andrew WiltshireBachelor of Engineering

(Electrical and Computer Engineering)1

st Class Honours

Doctor of Philosophy

Centre of Energy and Resource ManagementSchool of Engineering Systems

Faculty of Built, Environment and EngineeringQueensland University of Technology

Brisbane, Australia.

2007

Analysis of Disturbances in Large

Interconnected Power Systems

A thesis submitted in partial fulfillment of the requirements for thedegree of


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Abstract

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Analysis of Disturbances in Large Interconnected PowerSystems

by Mr. Richard Andrew Wiltshire

Principal Supervisor: Associate Professor Peter O’SheaSchool of Engineering Systems


Associate Supervisor: Professor Gerard LedwichSchool of Engineering Systems


Associate Supervisor: Dr Edward PalmerSchool of Engineering Systems


Abstract

World economies increasingly demand reliable and economical power

supply and distribution. To achieve this aim the majority of power

systems are becoming interconnected, with several power utilitiessupplying the one large network. One problem that occurs in a large

interconnected power system is the regular occurrence of system

disturbances which can result in the creation of intra-area oscillating

modes. These modes can be regarded as the transient responses of the

power system to excitation, which are generally characterised as decaying

sinusoids. For a power system operating ideally these transient responses

would ideally would have a “ring-down” time of 10-15 seconds.

Sometimes equipment failures disturb the ideal operation of powersystems and oscillating modes with ring-down times greater than 15

seconds arise. The larger settling times associated with such “poorly

damped” modes cause substantial power flows between generation nodes,

resulting in significant physical stresses on the power distribution system.


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Abstract

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If these modes are not just poorly damped but “negatively damped”,

catastrophic failures of the system can occur.

To ensure system stability and security of large power systems, the

potentially dangerous oscillating modes generated from disturbances

(such as equipment failure) must be quickly identified. The power utility

must then apply appropriate damping control strategies.

In power system monitoring there exist two facets of critical interest. The

first is the estimation of modal parameters for a power system in normal,

stable, operation. The second is the rapid detection of any substantial

changes to this normal, stable operation (because of equipment

breakdown for example). Most work to date has concentrated on the first

of these two facets, i.e. on modal parameter estimation. Numerous modal

parameter estimation techniques have been proposed and implemented,

but all have limitations [1-13]. One of the key limitations of all existing

parameter estimation methods is the fact that they require very long data

records to provide accurate parameter estimates. This is a particularly

significant problem after a sudden detrimental change in damping. One

simply cannot afford to wait long enough to collect the large amounts of

data required for existing parameter estimators. Motivated by this gap in

the current body of knowledge and practice, the research reported in this

thesis focuses heavily on rapid detection of changes (i.e. on the second

facet mentioned above).

This thesis reports on a number of new algorithms which can rapidly flag

whether or not there has been a detrimental change to a stable operating

system. It will be seen that the new algorithms enable sudden modal

changes to be detected within quite short time frames (typically about 1

minute), using data from power systems in normal operation.

The new methods reported in this thesis are summarised below.


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Abstract

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The Energy Based Detector (EBD): The rationale for this method is that

the modal disturbance energy is greater for lightly damped modes than it

is for heavily damped modes (because the latter decay more rapidly).Sudden changes in modal energy, then, imply sudden changes in modal

damping. Because the method relies on data from power systems in

normal operation, the modal disturbances are random. Accordingly, the

disturbance energy is modelled as a random process (with the parameters

of the model being determined from the power system under

consideration). A threshold is then set based on the statistical model. The

energy method is very simple to implement and is computationally

efficient. It is, however, only able to determine whether or not a suddenmodal deterioration has occurred; it cannot identify which mode has

deteriorated. For this reason the method is particularly well suited to

smaller interconnected power systems that involve only a single mode.

Optimal Individual Mode Detector (OIMD): As discussed in the previous

paragraph, the energy detector can only determine whether or not a

change has occurred; it cannot flag which mode is responsible for the

deterioration. The OIMD seeks to address this shortcoming. It uses

optimal detection theory to test for sudden changes in individual modes.

In practice, one can have an OIMD operating for all modes within a

system, so that changes in any of the modes can be detected. Like the

energy detector, the OIMD is based on a statistical model and a

subsequently derived threshold test.

The Kalman Innovation Detector (KID): This detector is an alternative to

the OIMD. Unlike the OIMD, however, it does not explicitly monitor

individual modes. Rather it relies on a key property of a Kalman filter,namely that the Kalman innovation (the difference between the estimated

and observed outputs) is white as long as the Kalman filter model is valid.

A Kalman filter model is set to represent a particular power system. If

some event in the power system (such as equipment failure) causes a


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Abstract

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sudden change to the power system, the Kalman model will no longer be

valid and the innovation will no longer be white. Furthermore, if there is a

detrimental system change, the innovation spectrum will display strong peaks in the spectrum at frequency locations associated with changes.

Hence the innovation spectrum can be monitored to both set-off an

“alarm” when a change occurs and to identify which modal frequency has

given rise to the change. The threshold for alarming is based on the

simple Chi-Squared PDF for a normalised white noise spectrum [14, 15].

While the method can identify the mode which has deteriorated, it does

not necessarily indicate whether there has been a frequency or damping

change. The PPM discussed next can monitor frequency changes and socan provide some discrimination in this regard.

The Polynomial Phase Method (PPM): In [16] the cubic phase (CP)

function was introduced as a tool for revealing frequency related spectral

changes. This thesis extends the cubic phase function to a generalised

class of polynomial phase functions which can reveal frequency related

spectral changes in power systems. A statistical analysis of the technique

is performed. When applied to power system analysis, the PPM can

provide knowledge of sudden shifts in frequency through both the new

frequency estimate and the polynomial phase coefficient information.

This knowledge can be then cross-referenced with other detection

methods to provide improved detection benchmarks.

Keywords

Power System Monitoring, Interconnected Power Systems, Power System Disturbances, Power System Stability, Signal Processing, Optimal

Detection Theory, Stochastic System Analysis, Kalman Filtering, Poly-

Phase Signal Analysis.


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Declaration

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Declaration

I hereby certify that the work embodied in this thesis is the result of

original research and has not been submitted for a higher degree

at any other University or Institution.

Richard Andrew Wiltshire

10 July 2007


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Table of Contents

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Table of Contents

Abstract ............................................................................................................iii

Keywords ......................................................................................................... vi

Declaration ......................................................................................................vii

Table of Contents ............................................................................................. ix

Table of Figures .............................................................................................. xv

List of Tables.................................................................................................. xxi

Acknowledgements......................................................................................xxiii

Dedication ..................................................................................................... xxv

Glossary.......................................................................................................xxvii

Chapter 1 ......................................................................................................... 29

1 Introduction ............................................................................................. 29

1.1 The Analysis of Large Interconnected Power Systems................... 29

1.2 The Monitoring of Australia's Large Interconnected Power System.

……………………………………………………………………..30

1.3 The use of Externally Sourced Simulated Data for Algorithm

Verification ................................................................................................. 31

1.4 Review of Existing Modal Estimation Methods ............................. 33


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1.4.1 Single Isolated Disturbance.....................................................33

1.4.1.1 Eigenanalysis of Disturbance Modes .................................. 33

1.4.1.2 Spectral Analysis using Prony’s Method ............................ 34

1.4.1.3 The Sliding Window Derivation .........................................36

1.4.2 Continuous Random Disturbances ..........................................38

1.4.2.1 Autocorrelation Methods..................................................... 38

1.4.2.2 Review of Kalman Filter Innovation Strategies ..................39

1.5 Review of Frequency Estimation Methods .....................................40

1.5.1 Polynomial-Phase Estimation Methods................................... 41

1.6 Conclusion.......................................................................................42

1.7 Organisation of the remainder of the thesis.....................................43

Chapter 2 .........................................................................................................45

2 Rapid Detection of Deteriorating Modal Damping.................................45

2.1 Introduction .....................................................................................45

2.2 The Power System Model in the Quiescent State ...........................46

2.3 The Power System Statistical Characterisation............................... 47

2.4 PDF Verification .............................................................................50

2.5 Setting the Threshold for Alarm...................................................... 52

2.6 Simulated Results ............................................................................52

2.7 Validation of Method using MudpackScripts ................................. 54

2.8 Application to Real Data ................................................................. 56

2.8.1 Results of Real Data Analysis .................................................58

2.9 Discussion .......................................................................................66


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2.10 Conclusion....................................................................................... 66

Chapter 3 ......................................................................................................... 69

3 Rapid Detection of Changes to Individual Modes in Multimodal Power

Systems ........................................................................................................... 69

3.1 Introduction..................................................................................... 69

3.2 The Stochastic Power System Model Revisited.............................. 70

3.3 Application of the Optimal Detection Strategy............................... 71

3.4 Individual Mode Detection Statistic Details ................................... 73

3.5 Statistical Characterisation of the Detection Statistic η .................. 74

3.6 Results ............................................................................................. 77

3.6.1 Simulated Results.................................................................... 78

3.7 Real Data Analysis .......................................................................... 82

3.8 Verification of Method.................................................................... 85

3.9 Real Data Analysis Results ............................................................. 90

3.10 Discussion ....................................................................................... 94

3.11 Conclusion....................................................................................... 95

Chapter 4 ......................................................................................................... 97

4 A Kalman Filtering Approach to Rapidly Detecting Modal Changes .... 97

4.1 Introduction..................................................................................... 97

4.2 Stochastic Power System Model..................................................... 98

4.3 The Kalman Application in Power System Analysis .................... 101

4.3.1 Kalman formulation .............................................................. 101


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4.3.2 State space representation of the power system model ......... 103

4.3.3 Kalman Solution.................................................................... 105

4.3.4 Detection using the Innovation..............................................106

4.4 Simulated Data Results ................................................................. 108

4.4.1 Simulation Type 1- damping change..................................... 111

4.4.2 Simulation Type 2- frequency change................................... 112

4.4.3 Simulation Type 3- damping and frequency change............. 113

4.4.4 Statistics of results................................................................. 114

4.5 Verification of the Kalman Method ..............................................115

4.6 Application to Real Data ............................................................... 116

4.6.1 Part I: Analysis of the Melbourne Data................................. 117

4.6.2 Part II: Combining multi-site data for enhanced SNR and

detection. ………………………………………………………………122

4.7 Guidance in tuning the Kalman Filter ...........................................126

4.8 Discussion on real data analysis ....................................................127

4.9 Conclusion.....................................................................................128

Chapter 5 .......................................................................................................129

5 A new class of multi-linear functions for polynomial phase signal

analysis ..........................................................................................................129

5.1 Introduction ...................................................................................129

5.2 The new class of multi-linear functions ........................................ 132

5.3 Designing new GMFC members...................................................134


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5.3.1 Algorithm for estimating the parameters of 4th order PPSs

based on ( )31 3,T n Ω .............................................................................. 136

5.4 Derivation of the Asymptotic Mean Squared Errors..................... 138

5.5 Simulations.................................................................................... 145

5.6 Application in Power System Monitoring..................................... 150

5.6.1 Real Data Analysis and Results ............................................ 153

5.6.2 Discussion on Real Data Analysis ........................................ 159

5.7 Conclusion..................................................................................... 160

Chapter 6 ....................................................................................................... 161

6 Discussion ............................................................................................. 161

6.1 Comparison of Proposed Detectors............................................... 166

Chapter 7 ....................................................................................................... 175

7 Conclusions and Future Directions ....................................................... 175

7.1 Conclusion..................................................................................... 175

7.2 Future Directions........................................................................... 176

Publications ................................................................................................... 179

Bibliography.................................................................................................. 181


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Table of Figures

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Table of Figures

Figure 1-1 States associated with the eastern Australian largeinterconnected power system (shaded). State capital cities thatrepresent generation nodes and measurement site location andratings are shown..............................................................................31

Figure 2-1 Model for quasi-continuous modal disturbances in a powersystem...............................................................................................46

Figure 2-2 Equivalent model for quasi-continuous modal oscillations in a power system....................................................................................47

Figure 2-3 Energy PDF and Histogram Comparison (60 second window)...........................................................................................................52

Figure 2-4 60 second Data Window of Energy Measurements with 1%False Alarm Rate Shown..................................................................53

Figure 2-5 Mode Trajectory of QNI Case13 MudpackScript Data..........55

Figure 2-6 Output Energy vs 1, 5, 10% thresholds..................................55

Figure 2-7 Short Term Energy Detection Applied to Real Data, reddenotes past data used to formulate long term estimates. ................57

Figure 2-8 24 hours of recorded angle measurements (2nd

October 2004),sites as indicated...............................................................................59

Figure 2-9 Queensland Estimated Impulse Response..............................59

Figure 2-10 New South Wales Estimated Impulse Response..................60

Figure 2-11 Victorian Estimated Impulse Response................................60

Figure 2-12 South Australian Estimated Impulse Response....................61

Figure 2-13 Queensland PDF Estimate with white noise verificationhistogram..........................................................................................61


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Table of Figures

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Figure 2-14 New South Wales PDF Estimate with white noiseverification histogram. ..................................................................... 62

Figure 2-15 Victorian PDF Estimate with white noise verificationhistogram.......................................................................................... 62

Figure 2-16 South Australian PDF Estimate with white noise verificationhistogram.......................................................................................... 63

Figure 2-17 Queensland 60 second energy measurements vs variousFARs shown..................................................................................... 63

Figure 2-18 New South Wales 60 second energy measurements vsvarious FARs shown........................................................................ 64

Figure 2-19 Victorian 60 second energy measurements vs various FARsshown. .............................................................................................. 64

Figure 2-20 South Australian 60 second energy measurements vs variousFARs shown..................................................................................... 65

Figure 3-1 Previously introduced stochastic power system model.......... 70

Figure 3-2 Generation of the optimal detection statistic.......................... 72

Figure 3-3 Mode 1 test statistic vs alarm threshold. ................................ 80

Figure 3-4 Mode 2 test statistic vs alarm threshold. ................................ 81

Figure 3-5 Spectral plot of mode contributions within system frequencyresponse............................................................................................ 82

Figure 3-6 Short Term Modal Detection Applied to Real Data...............84

Figure 3-7 Case13 Modal Damping and Frequency Trajectory. .............85

Figure 3-8 Spectral Estimate of Site Magnitude Response. ....................86

Figure 3-9 Estimates of Individual Modal Spectral Contributions -Brisbane (QNI).................................................................................87


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Figure 3-10 Estimates of Individual Modal Spectral Contributions -Sydney (NSW). ................................................................................87

Figure 3-11 Estimates of Individual Modal Spectral Contributions -Adelaide (SA)...................................................................................88

Figure 3-12 Individual Mode Monitoring - Mode 1 Brisbane. ................88

Figure 3-13 Individual Mode Monitoring - Mode 1 Sydney. ..................89

Figure 3-14 Individual Mode Monitoring - Mode 1 Adelaide.................89

Figure 3-15 Brisbane Mode 1 Test Statistic vs Time (1% FAR).............91

Figure 3-16 Brisbane Mode 2 Test Statistic vs Time (1% FAR).............92

Figure 3-17 Sydney Mode 1 Test Statistic vs Time (1% FAR). ..............93

Figure 3-18 Sydney Mode 2 Test Statistic vs Time (1% FAR). ..............93

Figure 3-19 Magnitude spectrum of voltage angles at different sites at24:00hrs............................................................................................94

Figure 4-1 Equivalent model for the individual response of a power

system to load changes. ....................................................................99

Figure 4-2 General Kalman filter estimator. ............................................99

Figure 4-3 Innovation PSD of: a) the 60 second interval prior to thedamping change, and b) the 60 second interval subsequent to thedamping change. ............................................................................111

Figure 4-4 Innovation PSD of: a) the 60 second interval prior to thefrequency shift, and b) the 60 second interval subsequent to thefrequency shift................................................................................112

Figure 4-5 Innovation PSD of: a) the 60 second interval prior to thedamping change, and b) the 60 second interval subsequent to thedamping and frequency change......................................................113


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Table of Figures

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Figure 4-6 Analysis of Normalised Innovation prior to suddendeteriorating damping at 120mins. ................................................115

Figure 4-7 Analysis of normalised innovation in the 60 seconds after thedeteriorating damping at 121mins. ................................................116

Figure 4-8 Melbourne frequency response estimate from LTE at 165minutes........................................................................................... 118

Figure 4-9 Comparison of (a) system output and (b) normalisedinnovation. .....................................................................................120

Figure 4-10 (a) Innovation sequence γ(n), (b) Innovation PSD at 196-197mins................................................................................................121



Figure 4-13 Normalised (a) Individual innovation PSDs for Sydney,Melbourne and Adelaide (b) Combination PSD at 196-197 mins.The new threshold corresponding to a 99.9% FAR....................... 125

Figure 4-14 Normalised (a) Individual PSD at 198-199 mins (b)Combination PSD at 198-199 mins showing new threshold for CI of99.9% FAR. ...................................................................................125

Figure 5-1 a4 estimate MSE vs. SNR for the final and intermediate parameter estimates........................................................................ 147




Figure 5-5 a0 estimate MSE vs. SNR for the final parameter estimates.149


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Figure 5-6 b0 estimate MSE vs. SNR for the final parameter estimates.........................................................................................................150

Figure 5-7 Measured Data, (a) Voltage Magnitude and (b) Phase. .......154

Figure 5-8 Reconstructed Signal ( ).r z t ...................................................155

Figure 5-9 Example of signal slice used for parameter estimation, down-sampled to 6.25Hz. Coloured signal shown is around the first phasedisturbance. ....................................................................................155

Figure 5-10 Second example of signal slice used for parameterestimation, down-sampled to 6.25Hz. Coloured signal shown isaround the second phase disturbance. ............................................156

Figure 5-11 b0 estimate. .........................................................................156

Figure 5-12 a0 estimate (phase deg).......................................................157

Figure 5-13 a1 estimate, 2ω π

= f . ............................................................157

Figure 5-14 a2 estimate (frequency rate), ω & . .........................................158

Figure 5-15 a3 estimate, ω && .....................................................................158

Figure 5-16 a4 estimate, ω &&& .....................................................................159

Figure 6-1 EBD outputs from three sites, NSW, VIC and SA...............169

Figure 6-2 OMID Mode 2 outputs from three sites, NSW, VIC and SA.........................................................................................................170

Figure 6-3 OMID Mode 1 outputs from three sites, NSW, VIC and SA.........................................................................................................171

Figure 6-4 Estimation of Spectral Mode Contributions from three sites, NSW, VIC and SA. ........................................................................172


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List of Tables

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List of Tables

Table 2-1 Relative Error of Moments ......................................................51

Table 2-2 Percentage of Alarms...............................................................54

Table 2-3 Percentage of False Alarms over initial 3 hours of Data.........65

Table 3-1 Qualitative Reference to Damping Performance (NEMMCO)*..........................................................................................................78

Table 3-2 Stationary Modal Parameters and Weights..............................79

Table 3-3 Damping Changes....................................................................80

Table 3-4 Alarms (1% FAR) ....................................................................81

Table 3-5 Long Term Modal Parameter Estimates ..................................85

Table 3-6 False Alarms (1% FAR) ..........................................................91

Table 4-1 Qualitative Reference to Damping Performance...................109

Table 4-2 Damping and Frequency Changes to Mode 1........................110

Table 4-3 Alarms (0.1% FAR)...............................................................114

Table 4-4 Damping and Frequency Long Term Estimates over 120-165mins................................................................................................118

Table 4-5 SNR Improvement through Combination of Site Analysis ...126

Table 5-1 Approximate Formulae for CRLBs ( )1 N >> [63].................139

Table 5-2 Parameter Values of 4th Order Polyphase Signal used inSimulations.....................................................................................146

Table 6-1 Comparison of detection method test statistics. ....................173


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Acknowledgments

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Acknowledgements

The author wishes to thank the following for their support during the

period of this research. Firstly I’d like to extend a sincere thank you to

Associate Professor Peter O’Shea, for his guidance, encouragement, input

and patience during the period of this research. Dr O’Shea has all the

qualities any candidate could ever ask for from a supervisor, both in his

technical expertise and his persona, thanks Peter.

I would also like to extend a warm thank you to Professor Gerard

Ledwich (QUT) for his comments, guidance, input and insightful

understanding of the research topic.

Other people I’d like to mention and thank are Dr Ed Palmer (QUT) for

his contributions, support and friendship over many years and Dr Chaun-

Li Zhang (QUT) for his endless supply of data when I needed it. I would

also like to thank Maree Farquharson (QUT) who, as fellow candidates,

supported and encouraged each other over our respective research

periods.

In addition I would like to extend a grateful thank you to David Bones

(NEMMCO) and David Vowles (University of Adelaide) for giving me

permission to use the MudpackScripts as an important validation tool

within this thesis, very much appreciated.

Finally, I would like to thank the following for the much welcome

financial support over the course of the research; Queensland University

of Technology for the APAI and QUTPRA scholarships, the QUT

Chancellery for providing the Vice-Chancellor’s Award, the QUT Faculty

of Built Environment and Engineering for the BEE financial top-up and

finally my current employer, CEA Technologies Pty. Ltd, for providing

me the study leave I required in the final stages of completing this body

of work.


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Dedication

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Dedication

There are a number of very special people I would like to dedicate this

work to; firstly my mother, Virginia Ann Bryan, who with her guidance,

support and never ending devotion has helped me mature into the man I

am today. I would also like to dedicate this work to my father, RogerWiltshire, for providing me the qualities that formulate a good engineer.

Also to my two boys, Peter and Jack, who I am eternally proud of and

encouraged by to strive to be a better father and person and finally to

three very special and lovely ladies, Catherine Louise Kowalski, Meg

Malaika (21/9/1966-19/3/2006) and Leslie Elizabeth Peters who in their

own extraordinary way have encouraged, inspired, taught and supported

me in my endeavours over the last few years.

Love to you all…


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Glossary

xxvii

Glossary

AESOPS Analysis of Essentially Spontaneous Oscillations in Power Systems

AMSE Asymptotic Mean Squared Error

ARI Adelaide Research and Innovation

CI Confidence Interval

COA Centre of Area

CRLB Cramér-Rao Lower Bounds

CP Cubic Phase Function

dB Decibels

DFT Discrete Fourier Transform

DTFT Discrete Time Fourier Transform

EBD Energy Based Detection

FAR False Alarm Rate

FFT Fast Fourier Transform

GMFC Generalized Multi-linear Function Class

GPS Global Positioning System

HAF Higher–Order Ambiguity Function

HP Higher-Order Phase Function

Hz Hertz

IF Instantaneous Frequency

IFR Instantaneous Frequency Rate

IIR Infinite Impulse Response

KID Kalman Innovation Detector

LT Long Term


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Glossary

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LTE Long Term Estimator

ML Maximum Likelihood

MSE Mean Squared Error

NEMMCO National Electricity Market Management Company Limited

NILS Non-linear Instantaneous Least Squares

NSW New South Wales

OIMD Optimal Individual Mode Detection

PDF Probability Density Function

PPM Polynomial Phase Method

PPS Polynomial Phase Signal

PSD Power Spectral Density

PWVD Polynomial Wigner-Ville Distributions

RMS (rms) Root Mean Squared

RV Random Variable

QLD Queensland

QP Quartic-Phase Function

QR Quadratic

QUT Queensland University of Technology, Brisbane, Australia

SA South Australia

SNR Signal to Noise Ratio

SVD Singular Value Decomposition

UA The University of Adelaide

VIC Victoria

W Watts


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Chapter 1

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Chapter 1

1 Introduction

1.1 The Analysis of Large Interconnected PowerSystems

The worldwide economic restructuring of the electrical utility industry

has formulated large interconnected distribution networks, resulting in a

greater emphasis on reliable and secure operations [17]. To ensure secure

and reliable operations, the large interconnected power systems require

ongoing wide-area observation and control. To meet these requirements

many wide-area monitoring methodologies have been proposed and

established [18-20]. One of the most well accepted approaches is to

monitor the power system at various locations within the distribution

network and to employ Global Positioning System (GPS) information to

synchronise the information acquired [21, 22]. With this approach, the

positioning of the measurement locations in the network is an important

issue which is discussed in [23].

Monitoring of power system stability is a critical issue for distributed

networks with a significant focus on the inter-area oscillations, whereby

this stability is largely dependent on all “inter-area oscillations” being

positively damped. The latter are oscillations that correspond to transient

power flows between clusters of generators or plants within a specific

area in the large interconnected power system [24]. Monitoring and

control of these oscillations is vitally important, and has proven far more


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Chapter 1

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difficult than monitoring and control of oscillations associated with a

single generator [24].

The inter-area oscillations (or modes) are damped sinusoids, at a given

frequency with a relevant damping factor. It is the “ring-down” time

associated with the damping factor that is of consequence in the transient

ability of the system to stabilise post disturbance. It is critical that the

transient time is short (and stable) to minimise power flows between the

generation clusters and minimise the associated stresses within the

generation/transmission infrastructure. As a consequence there has been

much work done in the area of damping factor estimation in large

distributed power systems. Previous estimation methods have employed

Eigen analysis [25-27] as well as Prony [28] analysis [29, 30]. For

accurate damping factor estimation, however, one typically requires large

amounts of data [12, 13]. Conventional damping factor estimation

techniques are therefore not suitable for rapidly detecting sudden modal

damping changes. This thesis addresses this shortcoming by presenting a

variety of new monitoring methods which are able to provide indications

of detrimental modal parameter change with short data records (typically

of the order of minutes).

1.2 The Monitoring of Australia's LargeInterconnected Power System.

In Australia, the power system associated with the eastern states is an

example of a large interconnected power system. The eastern Australian

distribution infrastructure contains a number of generation clusters andthere are inter-area modal oscillations which arise from the interaction of

these clusters. A generalised map of the cluster location in eastern

Australia is shown in Figure 1-1, with the capital cities representing the

generation nodes. Also listed in Figure 1-1 are the locations of the GPS


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Chapter 1

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monitoring sites as presented in [22]. Due to the importance of

monitoring inter-area oscillations, a number of partners have entered into

collaboration to effect the monitoring. These partners include QueenslandUniversity of Technology (QUT), and various transmission distributors as

listed in [22]. The wide-area GPS synchronised techniques outlined in

[22] and [23] have provided the real system data analysed in this thesis.

Queensland

Victoria

New South Wales

South Australia

Melbourne

SydneyAdelaide

Brisbane

275 kV

220 kV

330 kV

275 kVSouth PineBrisbane

Para Adelaide

RowvilleMelbourne

Sydney WestSydney

GPS Measurement Location& bus rating

City

275 kV

220 kV

330 kV

275 kVSouth PineBrisbane

Para Adelaide

RowvilleMelbourne

Sydney WestSydney

GPS Measurement Location& bus rating

City

Figure 1-1 States associated with the eastern Australian large interconnected power

system (shaded). State capital cities that represent generation nodes and

measurement site location and ratings are shown.

(Template image of Australia sourced from http://www.rrb.com.au/Images/Australia)

1.3 The use of Externally Sourced Simulated Datafor Algorithm Verification

The ultimate goal of power system monitoring algorithms is to perform

reliably in real power system scenarios. Before this can be achieved,


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Chapter 1

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however, the algorithms need to be tested via simulations. The simulation

environment allows conditions to be varied and consequent performance

to be evaluated in a controlled manner. For the simulation environment to provide useful testing, however, it must incorporate modelling which is

representative of real systems. Good modelling strategies help to verify

new techniques prior to real data implementation and provide confidence

in real data analysis results. The task of creating satisfactory models of

large dynamically interconnected systems is a challenging and non-trivial

task. Fortunately, for the purposes of this PhD research, the author was

given access to externally1 created simulation models and data, based on

the eastern Australian interconnected power system. The modelling of thesystem was performed by Adelaide Research & Innovation (ARI), a

group associated with The University of Adelaide (UA), Australia. The

formulation of the power system model and associated data was

commissioned and contracted by the National Electricity Market

Management Company Limited (NEMMCO) to provide benchmark

testing of modal estimation methods from various research centres. The

Centre of Energy and Resource Management within the School of

Engineering Systems at Queensland University of Technology was one of

the research centres that was benchmark tested in 2004 [31]. The

simulated data provided in [31] was referred to as the “ MudpackScripts”

by the University of Adelaide authors. It consisted of various non-

stationary data sets that were of interest in this thesis. The data set of most

interest for this thesis is MudpackScript “Case13” which contains large

detrimental step changes of damping. The details of the changes in the

MudpackScripts will be presented in Chapter 2 Section 2.7 where it is

first used for technique verification prior to real data analysis.

1 Externally in this context means not associated with Queensland University ofTechnology, Brisbane Australia.


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1.4 Review of Existing Modal Estimation Methods

With power systems becoming increasingly large and interconnected, the

resulting advantages in efficiency have been offset somewhat by thedisadvantage of greater vulnerability to system instability. This latter

problem has made it very important to be able to perform reliable

detection of system disturbances from modal oscillation data records.

These data records can be associated with either a single isolated

disturbance or with continuous random disturbances. The power

industry’s chief concern is in the detection of exponentially growing

disturbance modes potentially present in the power system. If a growing

mode is detected then the power utilities must introduce some dampening

to counteract the mode. As a result, methods for fast, reliable and accurate

estimation of the modal parameters are very important.

1.4.1 Single Isolated Disturbance

The parameter estimation methods in this section of the literature review

focus on the power system’s response to a single isolated disturbance.

1.4.1.1 Eigenanalysis of Disturbance Modes

There are a number of well established methods that have been used for

the analysis of power systems. Many of these methods assume that the

intrinsically non-linear power system can be approximated as a linear

system for small perturbations from the steady state. Under this

assumption Kundur et. al . [32] showed that eigenvalue analysis

techniques could be quite effective. Conventionally, eigenanalysis of a

power system is carried out by explicitly forming the system matrix, then

using the standard QR algorithm to compute the eigenvalues of the matrix

[32]. Modal oscillation parameters were then obtained from the

eigenvalues. This basic method has proven to be generally reliable and

has been extensively used by power utilities worldwide. Unfortunately


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this method is unsuitable for large interconnected systems [26]. To enable

large system mode monitoring the basic Eigen methods above have

undergone various adaptations. Byerly et. al. [7] developed the best-known algorithm – AESOPS (Analysis of Essentially Spontaneous

Oscillations in Power Systems). The advantage of AESOPS is that it does

not require the explicit formation of the system state matrix [7]. The

shortfall of the AESOPS method is the inadequacy for analysing very

large interconnected systems.

Uchida and Nagao [25] made another development in eigenanalysis by

proposing the use of the “ S matrix method”. In this method, it is

assumed that the dynamics of power systems can be linearly

approximated with a set of differential equations of the form, Ax=& ,

where is the (vector) state of the system and A is the system matrix.

The S matrix method transforms the matrix, A , into the matrix,

1( )( )S A hI A hI −= + − , where I is the unit matrix and h is a positive real

number. It can be shown that the dominant eigenvalues of S are the same

as the dominant eigenvalues of A , but with an appropriate choice for h,

can be computed with better numerical precision and speed [25]. The

refined Lanczos process is also employed to make high-speed calculation

possible [25]. Despite the computational advantage of the “ S matrix

method” eigenanalysis has limited application for very large

interconnected power systems [33].

1.4.1.2 Spectral Analysis using Prony’s Method

The spectral analysis of modal parameters for power system disturbance

monitoring is another area of research which has received much attention.

In this approach power system disturbance data records are spectrally

analysed immediately after a fault or disturbance. One popular technique

used for the spectral analysis is Prony’s method, which originated in an

earlier century [28]. Its ability to be practically implemented, though, was


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delayed until the advent of the digital computer. Numerical conditioning

enhancements developed in 1982 by Kumaresan and Tufts [3] broadened

its applicability to modal estimation. C.E Grund et. al. [33] comparedProny analysis to eigenanalysis and stated that “ Prony analysis has an

important advantage over eigenanalysis techniques in that it does not

require the derivation of a medium-scale model ”. Additionally, it can also

be applied to field measurements for the derivation of control design

models [33]. Many papers have been written on the use of Prony analysis

for oscillation modal parameter estimation [34], [29] , [35], with each

providing its own insights.

It should be noted however that the above applications of Prony’s method

assume that the signal contains little noise. In practice methods based on

the Prony technique are only effective where the noise power is relatively

small. Trudnowski et. al. [30] alluded to this when they stated for Prony

analysis “…The accuracy of the mode estimates is limited by the noise

content always found in field measured signals…”.

The poor conditioning of Prony’s method exists because of an ill-

conditioned matrix inversion in the method. To improve the ill

conditioning, Kumaresan and Tufts [3] [4] proposed using a “Pseudo-

Inverse” matrix, incorporating Singular Value Decomposition (SVD).

This technique was further explored by Kumaresan and Tufts in [4] and

was an improvement of the backward linear prediction methods proposed

by Nuttall [36], which in turn were improvements of Prony’s original

method.

This process of applying a truncated SVD analysis effectively increases

the SNR in the data prior to obtaining the solution vector. In [3],

simulations show that this method gives much more accurate estimates of

the modal parameters than traditional Prony methods. In [2], Kumaresan

also provided further enhancement to Prony’s method with the

introduction of FIR pre-filtering to reduce the sensitivity of measurement


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errors in observed signal samples when determining the parameters of

sinusoidal signals.

Gomez Martin and Carrion Perez also introduced some extensions in

working with noisy data with the application of Prony’s method [6] by

using a moving window in both forward and backward directions. This

application of forward and backward methodologies were further

explored by Kannan and Kundu [37].

A time-varying Prony method for instantaneous frequency estimation

from low SNR data was introduced by Beex and Shan [38]. This was

pertinent since power systems do have substantially non-stationary

components on occasions.

1.4.1.3 The Sliding Window Derivation

Prony’s method is a parametric spectral analysis method. Some authors

have pursued solutions using classical spectral analysis based on Fourier

methods. For example; Poon and Lee [39] developed a technique to

determine the modal parameters by employing a Sliding Window Fourier

Transform. The frequency components of the modes were first identified

in the frequency spectrum. The damping constants could then be obtained

by comparing the spectral magnitudes of a given modal component in

different time windows. These Fourier techniques proved to be quite

robust to noise and worked well as long as the oscillation modes were

well separated and could be separately distinguished within the Fourier

spectral domain.

Basically the method developed by Poon and Lee uses the rate of decay

of the Fourier Transform as a rectangular window slides to determine thedamping factor of the mode. The results of this method provided good

correlation compared to conventional techniques. However the

fundamental limitation of the Poon and Lee method was the tight

restrictions on the length of the window that could be used.


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It was subsequently shown that this restriction was needed to avoid errors

due to the interference from the superposition of the positive and negative

frequency components [9]. This interference was formulated by the largeside lobes of the spectral sinc function introduced by the rectangular

windowing. Hence, Poon and Lee specified that the window lengths only

have certain discrete values, at which the interference (zeros in the sinc

function) turned out to be zero. The problem was that the window length

was dependent upon the modal frequency, and hence this frequency had

to be accurately estimated prior to the windowing implementation.

Additionally, a different set of windows was required to process every

different mode present. O’Shea [8] extended Poon and Lee’s Fouriermethod and showed that a relaxation of the restriction on window lengths

could be achieved by applying a smooth tapering window (Kaiser

window) rather than a rectangular one [40], [9].

Although the sliding spectral window methods in [39] and [9] were robust

to noise, they only allowed analysis of multiple modes if the modes were

sufficiently well separated to be resolved with conventional Fourier

techniques. To deal with multiple closely spaced modes Poon and Lee

[41] developed a modified technique. For lightly damped closely spaced

low frequency oscillation modes exhibiting beat phenomenon they made

use of the imaginary part of the Fourier Transform of the swing curves.

Their simple dual modal case was modelled by:

1 2

1 1 2 2( ) cos(2 ) cos(2 )t t f t a f t e a f t eσ σ π π − −= +

(1.1)

and 1 f and 2 f were assumed to be close in frequency and unable to be

resolved using Fourier techniques.

Although not specifically stated by Poon and Lee, a major problem with

their method was that it was not straightforward to determine either 1 f or

2 f . It was therefore not straightforward to determine the damping factors.


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O’Shea [42] showed that a simpler and more reliable method of

determining modal parameters for closely spaced modes was to extend

the earlier Sliding Window method in [9] by calculating the spectrum inmore than two windows. Using the results from the multiple spectral

windows, a set of simultaneous equations in the desired parameters could

be created. These parameters were the complex amplitudes, frequencies

and damping factors of the modes. These simultaneous equations could

then be solved in a least squares sense [4] to obtain estimates for the

modal parameters. The presented simulations indicated good results.

1.4.2 Continuous Random Disturbances

The methods for modal analysis so far discussed all assumed that the data

record could be well modelled as a sum of complex exponential modes.

This is an acceptable model if the record has been obtained after a single

isolated disturbance. However this is not acceptable for a record obtained

from continuous random disturbances (which is the scenario for power

systems in normal operation [11]). The following sections investigate

modal parameter analysis in relation to continuous random disturbances.

1.4.2.1 Autocorrelation Methods

The estimation of modal parameters from data records corresponding to

continuous random disturbances was discussed by Ledwich and Palmer in

[11]. They reasoned that the continuous random disturbances exciting a

power system in normal operation should be fractal in nature, having a 1/ f

shaped spectrum [11], i.e. it should be equivalent to integrated white

noise. They also reasoned that a power system could be approximated as

an IIR filter. With these assumptions about the excitation and power

system, Ledwich and Palmer showed that if one differentiated the output

of the power system, the result would be equivalent to the output of an

IIR filter driven by white noise. Since the autocorrelation function of a

system driven by white noise reveals the impulse response of that system


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[43], then the autocorrelation function of the differentiated power system

disturbance output should be the impulse response of the power system,

i.e. it will have the form of a sum of complex exponentials. The modal parameters can then be determined using Prony analysis [11].

Autocorrelation techniques were further examined by Banejad and

Ledwich [12] to determine resonant frequencies and mode shape by

modelling disturbances using white noise to represent customer load

variations and an impulse to represent a disturbance. The simulation

results provided tangible relationships to a known system’s eigenvalue,

resonant frequencies and mode shape, but did not make any reference to

the limitations of mode spacing.

1.4.2.2 Review of Kalman Filter Innovation Strategies

In the 1950s, increased control requirements for advancing avionics led to

the formulation of what is now commonly known as the Kalman filter.

Although earlier radar tracking work by Swerling had formulated very

similar algorithms [44] the more highly recognised publications by

Kalman [45], [46] then Kalman and Bucy [47] were generally recognised

as the origins of the Kalman filter. Since that time the Kalman filter has

been recognised as a very important (and optimal) linear estimator; it has

been used extensively in a multitude of areas that encompass stochastic

models, state and parameter estimation and control requirements. There

have also been a multitude of Kalman filter variations for non-linear

systems, such as the extended Kalman filter [48] and unscented Kalman

filter [49]. In this thesis, the focus of interest is on the Kalman filter

innovation. The innovation is defined as the difference between the

measured output and the estimated output [50]. It is well known that the

innovation from a Kalman filter is spectrally white as long as the assumed

model parameters are valid [50, 51]. However under faulty or changed

conditions the innovation sequence will demonstrate large systematic

trends as the model will no longer represent the physical system [50].


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Various Kalman filter innovation approaches, that target fault detection,

diagnosis of dynamic systems and least squares estimation, are presented

in [50-55]. In this thesis the Kalman filter model is used to estimate thesystem output. By monitoring the whiteness of the innovation one can

detect if there are any sudden detrimental changes in the model

parameters [50], otherwise the innovation sequence is equivalent to the

original excitation under normal plant conditions [51].

1.5 Review of Frequency Estimation Methods

Although modal damping estimates are of critical importance, the

frequency of the modes also provides an opportunity to determine

changes in system behaviour and dynamics. In the case of large

interconnected power systems, if a particular major site disconnects from

a national grid (example South Australia disconnects from the Australian

Eastern network) then the resulting power system will undergo a dynamic

shift in an attempt to re-establish an equilibrium state. In doing so it is

expected that the frequencies of the remaining modes will also change.

Therefore to rapidly detect and estimate the changes in modal frequencies

is of critical importance.

The nature of the frequency changes that occur in a power system over

time will not be known precisely. To allow for the arbitrary nature of the

frequency trajectories, polynomial modelling will be used. Note that if the

frequency trajectory is a polynomial as a function of time, then the phase

trajectory will also be polynomial. It will be assumed that the

component/mode in question is given by:

( ) ( )0( ) , j nr w z n b e z nφ = + (1.2)

where b0 is the amplitude, the polynomial phase coefficients are given by

{ }0 1, , , P a a aK and the phase is:


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0

( ) P

p p

p

n a nφ =

= ∑ (1.3)

and ( )w z n is complex white Gaussian noise.

1.5.1 Polynomial-Phase Estimation Methods

To determine the frequency/phase trajectory of a component/mode

conforming to the model in (1.2) it is necessary to determine the

polynomial phase coefficients. An obvious solution to the problem of

obtaining estimates of these parameters is to use the direct Maximum

Likelihood (ML) method. However as discussed in [56] the

implementation of this method is very computationally intensive,

requiring a P -dimensional search. To overcome the computationally

restrictive implementation of the ML method, various authors have

introduced alternative strategies [57], [58], [59], [60] and [61].

Fundamentally these strategies employ multi-linear transforms that

reduce the search requirements from a P -dimensional search to a far more

computationally efficient P -one-dimensional search.

More recently O’Shea introduced a “time-frequency rate” representation

which is defined in equation (1.4) [16]:

( )( )

( )2

1 / 2

0

( , ) .r

N j m

z r r m

CP n z n m z n m e−

− Ω

=Ω = + −∑

(1.4)

This representation reveals the rate of change of frequency of a signal as a

function of time. This has some relevance to the power system scenario

where frequency changes are of particular interest. If n is set equal to 0 in

equation (1.4) the Cubic Phase (CP) function is obtained. This function

has been demonstrated to be very effective in the estimation of

Polynomial Phase Signals (PPS) up to orders of 3 P = . The

computationally efficient implementation of the CP was procedurally

outlined by O’Shea in [62]. This fast algorithm was shown to produce


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predictable estimate Mean Squared Errors (MSE) in close proximity to

relevant Cramér-Rao Lower Bounds (CRLBs), above a given Signal to

Noise Ratio (SNR) threshold. The relevant CRLBs associated with thePolynomial Phase Coefficients for a given order PPS are outlined in [63].

Generalised higher order (HP) phase functions were also introduced by

O’Shea in [62]. The HP function formed a multi-linear extension to the

CP function, specifically for the purposes of parameter estimation of PPS

of order greater than three. The definition of this function is shown below

in (1.5),

( )

2

( , ) , ,r r

P P j m

z z m

HP n K n m e− ΩΩ =∑ (1.5)

where ( ) ( ) ( )1,i

i i

r

r k k P I

z i r i r i K n m z n c m z n c m∗

= = Π + −

is the kernel and [ ]

. ir ∗

indicates conjugation of [ ]. iff 1ir = .

The parameters , ,i ic k , andir I are selected to ensure unbiased parameter

estimates for a phase polynomial of order P . The choice of these

parameters was conducted in a comparable manner to procedures outlined

in [61] and [59], to ensure unbiased phase coefficient estimates.

Further work on the HP function was explored by Farquharson et. al. in

[56]. New HP functions were devised that allowed parameters to be

determined in isolation [56]. It was noted in [56] that the new HP

functions had some similarities to functions introduced in [60], but they

also had some notable differences. In particular the new HP function

based estimates in [56] had much lower SNR thresholds and were

therefore much more practically applicable.

1.6 Conclusion

Despite the advances of the last decades for modal parameter estimation

techniques, there is a common recognition by many authors that no


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individual technique accounts for all the various situations that arise in

practice. Each tool has its own merits and applications, and provides a

different view into dynamic system behaviour. With technologicaladvances constantly changing the face of power systems, the need to

continually improve oscillation modal estimation algorithms has been

widely accepted as being very important. Reliable detection of sudden

detrimental changes in modal oscillations is also extremely important so

that catastrophic failures can be avoided. Relatively little work has been

done to date on optimal procedures for such detection.

In relation to frequency and frequency rate estimates methods outlined

earlier, it is important to keep in mind the desire of the power industry to

obtain estimates as quickly as possible with an acceptable level of error.

In the situation of angle measurements from sites around a national power

system, it is generally recognised that these recorded measurements have

a reasonable SNR. These SNRs should be adequate for the PPS modelling

approaches considered in this thesis for estimating/detecting changes in

post-separation modal frequencies.

Therefore the major focus of this thesis will be the rapid acquisition of

system information (both modal damping and modal frequency) under the

scenario of sudden detrimental change to a quasi-stationary large

interconnected power system.

1.7 Organisation of the remainder of the thesis

The remainder of the thesis is organised as follows. Chapter 2 introduces

a unique energy based method that is primarily focused on the rapid

detection of deteriorating modal damping in power systems. In Chapter 3

the technique in Chapter 2 is further extended and optimised for

monitoring of individual modal damping changes in multi-modal power

systems. Chapter 4 will then introduce another prospect of modal

parameter monitoring which is based around the Kalman filter innovation


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spectrum. Chapter 5 will introduce a new class of multi-linear functions

for polynomial phase signal analysis and examine implementation

opportunities in power system monitoring. Chapter 6 will be devoted to ageneral discussion and Chapter 7 will present conclusions and future

directions.


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Chapter 2

2 Rapid Detection of Deteriorating ModalDamping

2.1 Introduction

In this chapter a new method is introduced that deals with the problem of

detecting sudden changes in the damping of inter-area modes in large

interconnected power systems. The motivation for focus on rapid

detection, rather than the more traditional modal parameter estimation, is

drawn from the verity that standard modal parameter estimation methods

require long data records to yield accurate estimates – typically an hour or

more of data is needed. This is too long to wait if there is a sudden andseriously problematical change of damping.

While accurate estimation of the modes requires long time scales,

detection of sudden deterioration from a known quiescent point can be

done in much shorter time scales (typically a minute). The “sudden

change” can be detected very easily via a sudden change in the energy of

the system modes. However to be able to make an informed decision on

whether a change has actually occurred a statistical characterisation of the

quiescent system energy must be established. Once the statistical

characterisation has been formulated the thresholds for rapid detection of

modal deterioration may be set that can provide an alarm benchmark with

a defined confidence level.


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ideally obtained according to the procedure in [23]. This procedure

maximises the lightly damped inter-area mode information and minimises

the effect of the more heavily damped local modes (that are less critical inregards to overall supply stability and control).

Power System Filter

h(n)Integrator

white noise,

w(n)γ(n)

Differentiate

y(n)

Power System Filter h(n)

white noise

w(n) y(n)

output

x(n)Power System Filter

h(n)Integrator

white noise,

w(n)γ(n)

Differentiate

y(n)

Power System Filter h(n)

white noise

w(n) y(n)

output

x(n)

Figure 2-2 Equivalent model for quasi-continuous modal oscillations in a power

system.

With the power system model established, a statistical characterisation of

the system energy will be formulated in the following section.

2.3 The Power System Statistical Characterisation

The rationale behind the method proposed in this chapter is that the

energy of the output, y(n), will remain stable unless either the power

system filter transfer function or the excitation noise level changes

suddenly. If the power system transfer function changes such that there is

less damping of the excitation noise, then the result is more energy within

the output signal. Alternatively if there is a sudden (and sustained)

increase in excitation energy there could be a fault. In either case an

alarm should be created so that appropriate investigation/control can be

implemented.

This section will apply the knowledge of the power system model

established in [11] to generate a statistical system characterisation. A

formula is derived for a probability density function (PDF) of the energy


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y(n) under quasi-stationary operating conditions. The statistical

characterisation of the energy y(n) enables a reliable threshold to be set so

alarms can be raised if the energy deviates too much from these quasi-stationary operating conditions. These operating conditions can be

determined by the types of techniques described in [11]. As the detection

condition is for large detrimental change then the False Alarm Rate

(FAR) for detection is usually set fairly low (1% or lower). Such a low

FAR facilitates the minimisation of unwanted false alarms. Once an alarm

has been triggered, further monitoring of the system energy is necessary.

Sequential data windows are collected and a statistical analysis with

respect to the PDF is undertaken. Repeated alarms due to consistentlyhigh energy readings would induce corrective action by the power system

utility.

To develop the PDF for the energy in y(n) we return to the model in

Figure 2-2. It follows from Figure 2-2 that the output signal’s discrete

Fourier transform is:

( ) ( ) ( ),Y k H k W k = (2.1)

where H(k) is the discrete Fourier transform (DFT) of h(n), W(k) is the

DFT of w(n) and Y(k) is the DFT of y(n). Now, according to Parseval’s

theorem, the energy of y(n) can be determined from the samples in either

the time domain or the frequency domain. Because the samples are

independent in the frequency domain, though, this domain is most

conducive to developing a statistical characterisation. Note also, that for

real signals, all the information in the frequency domain is contained in

the positive half of the spectrum – the information in the negative half is

just a copy of that contained in the positive half. Using Parseval’s

theorem and the fact that half the energy is contained in the positive half

of the spectrum for real signals, total energy of y(n) is:


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49

}|)0(||)0(|2

1|)(||)({|

2 222/

1

22W H k W k H

N E

N

k

∑=

+= (2.2)

Note that ( )W k is a complex Random Variable (RV) with real and

imaginary parts:

( ) ( ){ } ( ){ }2 2 2

Re ImW k W k W k = + (2.3)

Now if the variance of w(n) is 2σ , then the left hand side of (2.3) is a chi-

squared RV with two degrees of freedom and variance, N

2σ [15].

Therefore the PDF at any discrete ensemble frequency ik is:

( ){ } 22 2 , xN

i

N f W k e σ

σ

−

= (2.4)

where x is the random variable power.

Using (2.1) and (2.4) the PDF of ( )ik Y can be deduced to be:

( )

( )

( )2 2

2 2

2 2

1

( ) ( )

.

i

i

y k w

i i

N

H k x

i

x f Y f

H k H k

N e

H k

σ

σ

−

=

=

(2.5)

From (2.2) it is evident that the energy is obtained by summing / 2 1 N +

RVs and then scaling by 2/N . Furthermore these RVs have PDFs given by

(2.5). The PDF of the sum is obtained by convolving the PDFs of all the

RVs being summed. That is, the PDF of this sum is:

( ) ( ) ( ) ( )/ 2 / 2 1 0/ 2 / 2 1 0 N N y y N y N y

f x f x f x f x− −= ∗ ∗ ∗L (2.6)


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( ) ( )

( ) ( )

( ) ( )

2 212

2 2

2 2

2 012

2 1

2

2 / 2

0

1 ,

/ 2

xN H

xN H

xN H N

H e

N H e

H N e

σ

σ

σ σ

− −

− −

− −

− −

− −

− −

= ∗

∗ ∗ L

(2.7)

where * denotes convolution.

Finally the PDF of the energy is obtained from the sum by scaling by 2/N .

The final energy PDF must therefore have its axes re-scaled accordingly.

From the PDF the threshold for detection of change can be formulated.

Establishing say the 1% false alarm rate is obtained via the cumulative

summation of the PDF area until the 99% point is determined.

2.4 PDF Verification

To verify the theoretically determined system output PDF, a comparison

of a theoretical PDF and simulated histogram was undertaken. Using

known modal parameters, simulations created a collection database of

outputs that were formulated into a histogram. A theoretical PDF was

then formulated and compared directly to the histogram.

The procedure for verification involved 10,000 simulation runs of random

noise, ( )~ 0,1 N , feeding a known modal system, depicted in Figure 2-2

and defined below:

( ) ( ) ( )1 2 ,h n h n h n= + (2.8)

where

( ) ( )sin 1,2ini i i ih n A e n iσ ω φ −= + = (2.9)

with modal parameters:

11 1 1 11.7 / , 0.4 , 1, 0r s s Aω σ φ

−= = − = = o


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12 2 2 22.7 / , 0.52 , 1, 0r s s Aω σ φ

−= = − = = o

Energy measurements were based on a 20, 40 and 60 second data window

with a sampling rate of 5Hz. Statistical characteristics of the simulated

histogram and of the theoretical PDF were calculated and compared. The

statistical characteristics examined were the first three central moments,

mean, variance and skewness [64]. The percentage errors between the

theoretical and simulated PDFs are shown in Table 2-1. The errors are all

comparatively low, inspiring confidence in the fact that the derived PDF

is correct.

TABLE 2-1 R ELATIVE ERROR OF MOMENTS

TIME WINDOW 20SEC 40SEC 60SEC

NOISE VARIANCE 1.0 1.0 1.0

MEAN 0.04% 0.43% 1.67%

VARIANCE 0.82% 1.65% 0.24%%

ERROR

SKEWNESS[64] 2.59% 5.76% 6.53%

Further validation of the theoretical PDF is provided by a visual

comparison with a histogram. The result from a 60sec analysis window is

shown in Figure 2-3. There is a close visual alignment.


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5 10 15 20 25

0

10

20

30

40

50

60

70

80

Output half energy hisogram vs PDF

Energy-Joules

Output Energy Histogram vs PDF

5 10 15 20 25

0

10

20

30

40

50

60

70

80

Output half energy hisogram vs PDF

Energy-Joules

Output Energy Histogram vs PDF

Figure 2-3 Energy PDF and Histogram Comparison (60 second window).

2.5 Setting the Threshold for Alarm

From the PDF the threshold for detection of change can be formulated.

Establishing say the 10% false alarm rate is via the cumulative

summation of the PDF area until the 90% point is determined.

2.6 Simulated Results

It will be seen in this section that the simulations for detecting change

provided good results. The modal values were initially set to:

Mode 1: ( )0.4 sin 2t e t − 1 0.4σ ∴ = − 1& 2.0 /r sω =

Mode 2: ( )0.52 sin 2.7t e t − 2 0.52σ ∴ = − 2& 2.7 /r sω =

In the 300 minutes of simulated data, mode 1 underwent two damping

changes; a step deterioration to 1 0.1σ = − damping at 100 mins and

another step change to 1 0.2σ = − damping at 200 mins. The variance of


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the driving noise, w(n), was set to unity. The output, y(n), was generated

as per the model shown in Figure 2-2. Energy measurements for the

output were then taken over 1 minute block periods. The first 100 minutesrepresent the quasi-stationary system prior to the damping changes.

The graphical results and relevant statistics are shown in Figure 2-4 and

Table 2-2 respectively. The results are as one would expect – a low alarm

rate for the quiescent conditions, a high alarm rate during the major

damping change, and a lesser alarm rate during the period when the

damping “corrects itself” somewhat.

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

Half Output Energy vs Time

Time-Minutes

E n e r g y - J o u l e s

Energy

1% FAR

Output Energy vs Time

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

Half Output Energy vs Time

Time-Minutes


Energy

1% FAR

Output Energy vs Time

Figure 2-4 60 second Data Window of Energy Measurements with 1% False Alarm

Rate Shown.


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TABLE 2-2 PERCENTAGE OF ALARMS

FALSE ALARM R ATE SET AT 1%

0-100 MIN NO CHANGE

11 4.0

−−= sσ

100-200 MIN

STEP CHANGE 1

1 1.0 −−= sσ

200-300 MIN STEP CHANGE

11 2.0

−−= sσ

% OF TIMEALARM O N

0.0% 80.0% 32.0%

2.7 Validation of Method using MudpackScripts

In this section the energy based method is verified using the Case13

MudpackScript data.

This data set is useful as it contains an initial 2 hour section that may be

regarded as representing the power system in a quasi-stationary state with

acceptable damping. It is this initial 2 hour section that is analysed by the

LTE to generate the required system PDF. From this the desired threshold

can be set. This data set used for verification represents the Queensland

(QNI) mode of the power system. Within this QNI data, the damping

suddenly deteriorates from an acceptable -0.25 to a very

undesirable 0.05− . The trajectory of the damping factor associated with

the Case13 script can be seen in Figure 2-5.

The detection results from the analysis of the first 12 hours of data can be

seen in Figure 2-6. The analysis window used is a block window set to 60

seconds at a 5Hz sampling rate. Comparison of Figure 2-5 and Figure 2-6

shows that the energy experiences sudden jumps when the damping

experiences sudden jumps, as one would hope. Various different alarm

thresholds are shown in Figure 2-6; for the data under analysis it is seen

that if one uses a 1% false alarm rate, one only gets alarms when there are

genuine damping changes.

In the following section, the energy detection technique will be applied to

real multi-site data obtained from wide area monitoring sites.


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2 4 6 8 10 12

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Time-hrs

D a m p i n g

MudpackScript Case13 QNI Damping Trajectory

Figure 2-5 Mode Trajectory of QNI Case13 MudpackScript Data.

0 2 4 6 8 10 120

1

2

3

4

5

6

7

x 10-3 Output Energy vs Time

Time-hrs


Energy

10% FAR

5% FAR

1% FAR

Figure 2-6 Output Energy vs 1, 5, 10% thresholds.


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2.8 Application to Real Data

To apply the energy detection method to a real data situation a long term

estimator is required to provide an estimation of the quiescent modalvalues. Initially the long term estimator establishes estimates for the

system transfer function. From this an estimated impulse response can be

determined. With this knowledge an approximation of the variance of the

excitation signal, w(n) in Figure 2-1, can be estimated. Once the system

transfer function, h(n), and the excitation variance, 2σ , have been

estimated then the expected energy PDF can be formulated. It is this

formulated PDF that enables a threshold to be set (given a suitable false

alarm rate). Note that the long-term estimate is periodically updated, and

the threshold is changed, based on the updated estimate.

The process for simultaneously performing the long-term estimation,

generating the energy estimate, thresholding and alarming is depicted

diagrammatically in Figure 2-7. As shown in Figure 2-7, the first task is

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